# Binary Operations on Arrays¶

Tensor

Einstein

Nutils

1

$$\mathbf{a} \in \mathbb{R}^n$$

$$\mathbf{b} \in \mathbb{R}^n$$

$$c = \mathbf{a} \cdot \mathbf{b} \in \mathbb{R}$$

$$c = a_i b_i$$

c = (a*b).sum(-1)

2

$$\mathbf{a} \in \mathbb{R}^n$$

$$\mathbf{b} \in \mathbb{R}^m$$

$$\mathbf{C} = \mathbf{a} \otimes \mathbf{b} \in \mathbb{R}^{n \times m}$$

$$C_{ij} = a_i b_j$$

C = a[:,_]*b[_,:]

C = function.outer(a,b)

3

$$\mathbf{A} \in \mathbb{R}^{m \times n}$$

$$\mathbf{b} \in \mathbb{R}^n$$

$$\mathbf{c} = \mathbf{A}\mathbf{b} \in \mathbb{R}^{m}$$

$$c_{i} = A_{ij} b_j$$

c = (A[:,:]*b[_,:]).sum(-1)

4

$$\mathbf{A} \in \mathbb{R}^{m \times n}$$

$$\mathbf{B} \in \mathbb{R}^{n \times p}$$

$$\mathbf{C} = \mathbf{A} \mathbf{B} \in \mathbb{R}^{m \times p}$$

$$c_{ij} = A_{ik} B_{kj}$$

c = (A[:,:,_]*B[_,:,:]).sum(-2)

5

$$\mathbf{A} \in \mathbb{R}^{m \times n}$$

$$\mathbf{B} \in \mathbb{R}^{p \times n}$$

$$\mathbf{C} = \mathbf{A} \mathbf{B}^T \in \mathbb{R}^{m \times p}$$

$$C_{ij} = A_{ik} B_{jk}$$

C = (A[:,_,:]*B[_,:,:]).sum(-1)

C = function.outer(A,B).sum(-1)

6

$$\mathbf{A} \in \mathbb{R}^{m \times n}$$

$$\mathbf{B} \in \mathbb{R}^{m \times n}$$

$$c = \mathbf{A} : \mathbf{B} \in \mathbb{R}$$

$$c = A_{ij} B_{ij}$$

c = (A*B).sum([-2,-1])

Notes:

1. In the above table the summation axes are numbered backward. For example, sum(-1) is used to sum over the last axis of an array. Although forward numbering is possible in many situations, backward numbering is generally preferred in Nutils code.

2. When a summation over multiple axes is performed (#6), these axes are to be listed. In the case of single-axis summations listing is optional (for example sum(-1) is equivalent to sum([-1])). The shorter notation sum(-1) is preferred.

3. When the number of dimensions of the two arguments of a binary operation mismatch, singleton axes are automatically prepended to the “shorter” argument. This property can be used to shorten notation. For example, #3 can be written as (A*b).sum(-1). To avoid ambiguities, in general, such abbreviations are discouraged.